Chapter 5 Integers

Chapter 5

Integers

"In the lower grades, students may have connected negative numbers in appropriate ways to informal knowledge derived from everyday experiences, such as below-zero winter temperatures or lost yards on football plays. In the middle grades, students should extend these initial understandings of integers. Positive and negative integers should be seen as useful for noting relative changes or values. Students can also appreciate the utility of negative integers when they work with equations whose solution requires them, such as 2x + 7 = 1."

--Principles and Standards for School Mathematics

"By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g., situations of owing money or measuring elevations above and below sea level), students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers make sense."

--Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

Many everyday situations cannot be adequately described without using both positive and negative numbers. Profit and loss, temperatures above and below 0?, elevations above and below sea level, and deposits and withdrawals are just a few examples. This chapter introduces negative numbers by extending your knowledge of whole numbers to the set of integers.

In Activities 1 and 3, represents a proton, a subatomic particle with a positive electrical charge of one unit, and ~ represents an electron, a particle with a negative electrical charge of one unit. Because protons and electrons have opposite charges, when a proton and an electron are paired together, they neutralize each other; that is, the pair has an electrical charge of zero. You will use concrete models, such as charged particles, to represent integers and your understanding of the operations with whole numbers to develop the concept of absolute value and the algorithms for the operations with integers.

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72 Chapter 5 ? Integers

Activity 1: Charged Particles

PURPOSE MATERIALS GROUPING GETTING STARTED

Examples:

Use the charged-particle model to represent integers and to explore absolute value.

Other: Two different-colored chips (or squares cut from tag board), 15 of each color

Work individually or in small groups.

Use the dark chips to represent protons and the light chips to represent electrons. Construct two different models that represent each integer and sketch your models in the boxes.

The set at the right shows one way to represent the number 2.

If the protons and electrons are paired, 2 protons are left over. The net electrical charge is 2.

The set at the right shows one way to represent the number -3.

If the protons and electrons are paired, 3 electrons are left over. The net electrical charge is -3.

5

5

1

1

2

2

0

0

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Chapter 5 ? Activity 1 73

1. Look back at your models on the previous page. If you have not already done so, represent each integer using the fewest number of protons or electrons possible and sketch the model in the space provided.

a. +5

b. -1

c. -2

d. 0

2. What is the fewest number of particles needed to model each integer in Exercise 1? a. ______ b. ______ c. ______ d. ______

The answers to Exercise 2 are the absolute values of the integers in Exercise 1. Since the absolute value of an integer is represented by the fewest number of protons or electrons, it will always be 0 or a positive number. Since the distance between two points is always a positive number or 0, the absolute value of an integer may also be defined as the distance from 0 to the point corresponding to the integer on a number line.

Examples: The absolute value of 7, written as 7 , is 7. -8 = 8.

3. What is the absolute value of each of the following integers?

a. -15 ____ b. 12 ____ c. 0 ____ d. -5 ____

4. Use your results from the preceding exercises to complete each statement.

a. The absolute value of a positive integer is ______

b. The absolute value of a negative integer is ______

c. The absolute value of 0 is ______

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74 Chapter 5 ? Integers

Activity 2: Coin Counters

PURPOSE MATERIALS GROUPING GETTING STARTED

Use a game to discover algorithms for integer addition.

Other: A paper cup, 10 pennies, and a game marker

Work in pairs or in groups of 2 or 3.

? At the beginning of the game, each player places a game marker on zero on a number line like the one below.

? Players alternate turns. ? On your turn, place six pennies in the cup, cover the opening with

your hand, shake the cup thoroughly, and drop the coins onto the table. Each HEAD means move your marker to the right one unit; each TAIL means move it to the left one unit. ? The first player to go past 10 or -10 is the winner. If there is no winner after ten turns, the player closest to 10 or 10 wins.

?10 ?9 ?8 ?7 ?6 ?5 ?4 ?3 ?2 ?1 0 1 2 3 4 5 6 7 8 9 10

Play the game twice. When you have finished, each player should answer the following questions.

1. Did you find a way to quickly determine where to place your marker after a coin toss? Explain.

2. If you were to represent the number of HEADS with an integer, would you use a positive or a negative integer?

3. Would you use a positive or a negative integer to represent the number of TAILS?

4. Did your marker ever end up an odd number of units away from where it was at the start of your turn? Explain.

5. At the end of a turn, did your marker ever end up in the same place where it started? Explain.

6. Use coins to construct two different representations for each integer. You may use more or less than six coins in a model.

a. 4

b. -3

c. 0

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Chapter 5 ? Activity 2 75

You have seen how coins can be used to represent integers. Coins can also be used to model addition of integers. Think of the HEADS as a positive integer and the TAILS as a negative integer. For example, tossing 2 HEADS and 4 TAILS is the same as adding 2 and 4.

HHT T T T

H H T T

T T

2 4

1. a. Why do the paired coins cancel each other out?

b. If you tossed this combination of coins, how would you move your marker?

c. What integer is represented by the combination of coins?

d. Complete the equation: 2 + (-4) =

.

2. Use coins to find the following sums. Make a sketch of your work. You may use more than six coins.

a. 1 + (-5)

b. 6 + (-4)

c. 3 + (-3)

d. -5 + (-2)

Use the coin model to answer the following questions. 3. a. Is the sum of two negative numbers positive or negative? b. How can you determine the sum of two negative numbers without using coins?

4. When is the sum of a positive and a negative number

a. equal to 0?

b. positive?

c. negative?

5. How can you determine the sum of a positive and a negative number without using coins?

6. Use your rules from Exercises 3 and 5 to compute the following:

a. -17 + 25

b. 13 + (-7)

c. -36 + (-19) d. -11 + 11

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